Sharp weighted estimates for dyadic shifts and the $A_2$ conjecture

Abstract

We give a self-contained proof of the $A_2$ conjecture, which claims that the norm of any Calderon-Zygmund operator is bounded by the first degree of the $A_2$ norm of the weight. The original proof of this result by the first author relied on a subtle and rather difficult reduction to a testing condition by the last three authors. Here we replace this reduction by a new weighted norm bound for dyadic shifts - linear in the $A_2$ norm of the weight and quadratic in the complexity of the shift -, which is based on a new quantitative two-weight inequality for the shifts. These sharp one- and two-weight bounds for dyadic shifts are the main new results of this paper. They are obtained by rethinking the corresponding previous results of Lacey-Petermichl-Reguera and Nazarov-Treil-Volberg. To complete the proof of the $A_2$ conjecture, we also provide a simple variant of the representation, already in the original proof, of an arbitrary Calderon-Zygmund operator as an average of random dyadic shifts and random dyadic paraproducts. This method of the representation amounts to the refinement of the techniques from nonhomogeneous Harmonic Analysis.

38 pages; v2: the weighted bound for shifts is now quadratic in shift complexity (as opposed to cubic in v1)

References in corpus (6)

• Sharp weighted estimates for classical operators
• Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures
• Linear bound for the dyadic paraproduct on weighted Lebesgue space $L_2(w)$
• The vector-valued non-homogeneous Tb theorem
• The sharp weighted bound for general Calderon-Zygmund operators
• Current-phase relation of graphene Josephson junctions

Cited by in corpus (13)

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• Sharp weighted bounds involving A_\infty
• Equivalent definitions of dyadic Muckenhoupt and Reverse Hölder classes in terms of Carleson sequences, weak classes, and comparability of dyadic $L\log L$ and $A_{\infty }$ constants
• Random "dyadic" lattice in geometrically doubling metric space and $A_2$ conjecture
• A note on the off-diagonal Muckenhoupt-Wheeden conjecture
• A Bellman function proof of the $L^2$ bump conjecture
• Regularizations of general singular integral operators
• Strong weighted and restricted weak weighted estimates of the square function
• A Note On ${\ell }^r$-Valued Calderon-Zygmund Operators
• An integral arising from dyadic average of Riesz transforms